Recalling (REF ) and (REF ), \(c_k^{p}r_k^N||u_k||_p^{N-p}\psi _k^{p-1}\rightarrow 0\)  in \(L^\infty _{\rm loc}(\mathbb {R}^N)\) .
Then using the same argument as in ([1]}, Section 3) or ([2]}, Lemma 17), we have by applying elliptic estimates to (REF ),
\(\varphi _k\rightarrow \varphi \,\,{\rm in}\,\, C^1_{\rm loc}(\mathbb {R}^N)\,\,{\rm as}\,\,k\rightarrow \infty ,\) 
